Vertical profiles of the local gravitational instability parameter, Q_3D, for discs with polytropic vertical density distributions assuming Q = 0.4, when γ = 5/3 (upper panel) or γ = 1 (lower panel). In each panel, the selected values of n are such that the distribution is convectively stable for the assumed γ. The case n = ∞ is the SGI, for which Q_3D(z) is analytic. The horizontal dashed line indicates the instability threshold, Q_3D = 1.